Limits of Integration and finding k

[scothoward]

Homework Statement

Joint pdf given as kxy for 0 < x < y < 1.

Find the value of k.

The Attempt at a Solution

I understand the process of finding k - doing the double integral and setting it to 1. What I don't understand is the limits of integration for y.

I've seen two different limits set, but I still cannot seem to figure out how and why it is done.

I have seen the integral of x from 0 to 1 and the integral of y from x to 1. I have also seen the integral of x from 0 to 1 and the integral of y from 0 to y. Both give the correct answer of k = 8. My question is how do you go about choosing the limits for the y integral?

Thanks a lot!

 

[Unco]

I have seen the integral of x from 0 to 1 and the integral of y from x to 1. I have also seen the integral of x from 0 to 1 and the integral of y from 0 to y.

The latter does not make sense; it more likely read "integral of y from 0 to 1 and the integral of x from 0 to y".

The region being integrated over is the triangle with vertices (0,0), (1,1) and (0,1). Using vertical strips, we have the limits

$$\int_{x \, = \, 0}^{x=1} \int_{y=x}^{y=1} kxy \, dy \, dx$$,

or, using horizontal strips, we have

$$\int_{y=0}^{y=1} \int_{x=0}^{x=y} kxy \, dx \, dy$$.

(The limits in the integrals read "x=0", "x=1" etc -- the equals signs look like minus signs in this latex, unfortunately; I've included them for clarity.)

Both are equivalent.